Subdivision Loop cut doesn't hold
Ok, well first, you might be doing it wrong...

When you make a loop cut, click to establish the loop cut. Once the loop cut has been made, drag it (not holding down the mouse) to the desired location and click again to set it.

Does that work?

Also, just so you know, you don’t ever want to have vertices overlapping—which leads me to:

Pro-tip: if you’re using subsurf and want sharp edges, hit N, which opens up a side bar on the right. In it, in the default Item tab, you’ll see Mean Crease. Select the edges you want to keep sharp, and set their Mean Crease value to 1.00 (or any value in between) to accomplish your creases with no extra vertex data and no overlapping vertices (which will come back to bite you).

Good luck, and happy Blendering,

John